Consider the i.i.d. Gaussian variables , where and . The random variables from a linear combination of , e.g.

are correlated. Their covariance is given by

as the covariance between two independent standard normal variables is zero.

Consider a -vector of independent standard normal variables , and define the -vector of random variables as the linear combination of the variables in and a -vector of constants :

then the covariance matrix of is and the mean is .

The p.d.f. of a univariate Gaussian variable is

while the p.d.f. of a multivariate Gaussian variable (-vector) is